Finding roots of a function is often a task which faces mathematicians. For simple functions, such as linear ones, the task is simple. When functions become more complex, such as with cubic and quadratic functions, mathematicians call upon more convoluted methods of finding roots. For many functions, there exist formulas which allow us to find roots. The most common such formula is, perhaps, the quadratic formula. When functions reach a degree of five and higher, a convenient, root-finding formula ceases to exist. Newton’s method is a tool used to find the roots of nearly any equation. Unlike the cubic and quadratic equations, Newton’s method – more accurately, the Newton-Raphson Method – can help to find roots of nearly any type of function, including all polynomial functions.
Newton’s method use derivative calculus to find the roots of a function or relation by first taking an approximation and then improving the accuracy of that approximation until the root is found. The idea behind the method is as follows. Given a point,
P(Xn,Yn), on a curve, a line tangent to the curve at P crosses the X axis at a point whose X coordinate is closer to the root than Xn. This X coordinate, we will call Xn+1. Repeating this process using Xn+1 in place of Xn will return a new Xn+1 which will be closer to the root. Eventually, our Xn will equal our Xn+1. When this is the case, we have found a root of the equation. This method may be unnecessarily complex when we are solving a quadratic or cubic equation. However, the Newton-Raphson Method compensates for its complexity in its breadth. The following examples show the versatility of the Newton Raphson Method.
Example 1 is a simple quadratic function. The most practical approach to finding the roots of this equation would be to use the quadratic equation or to factor the polynomial. However, the Nowton-Raphson method still works and allows us to find the roots of the equation. The initial number, Xn, 3, is a relatively poor approximation. The choice of 3 illustrates that the initial guess can be any number. However, as the initial approximation worsens, the calculation becomes more laborious.
Example 2 demonstrates one of the advantages to Newton’s method. Function 2 is a Quintic function. Mathematician, Niels Henrik Abels proved that there exists no convenient equation, such as the cubic equation, which can help us find the function’s roots.
[-2, -1], [0, 1] and [1, 2]. Looking at the root in the interval [1,
Section 10.1 of the Algebra 1 textbook analysis is performed in the context of a specific classroom, students, reading proficiency, and learning goals. In essence textbook is evaluated from reader’s perspective and the learning of complex and abstract mathematical models. Chapter 10 objective is to develop foundation to graph and solve quadratic equations (Larson, Boswell, Kanold & Stiff, 2007). Applicable California Common Core Content Standards for Mathematics are moderately vigor and requires students to:
If you are asked to give an exact solution for a quadratic equation that does not have x-intercepts, then you will answer that question using the variable i. Say you need to find the square root of negative sixty-four. We know that the perfect square of positive sixty-four is eight. What we are going to do is take out i, resulting in the square root of positive sixty-four, which we know is eight, therefore the answer is plus or minus eight i. It is plus or minus because square roots can be positive or negative because a negative times a negative is also equal to a positive. You can also simplify an equation if there is a constant before the negative under the radical. You do the same thing in terms of simplifying as you would do without the constant. After you get your imaginary number you put the constant in the correct position, and then you are left with a complex number, such as a+bi. You can also simplify i when it is involved in a polynomial. If you multiply out two polynomials that have imaginary numbers in them you may end up with i with an exponent attached. You can use your previous knowledge of patterns to simplify the equation. Say you end up with a term along the lines of thirty-six i squared. We know that i to the second power is the same as negative one. From there we can multiply negative one and thirty-six to result in the product of negative thirty-six. As you have
Ball, Rouse. “Sir Isaac Newton.” A Short Account of the History of Mathematics. 4th ed. Print.
Rubba, J. (1997, February 3). Ebonics: Q & A. Retrieved July 12, 2010, from http://www.cla.calpoly.edu/~jrubba/ebonics.html
The value for D turns out to be zero for all values of the cubic polynomials thereby preventing us to form a conjecture for D = 0
Kinematics unlike Newton’s three laws is the study of the motion of objects. The “Kinematic Equations” all have four variables.These equations can help us understand and predict an object’s motion. The four equations use the following variables; displacement of the object, the time the object was moving, the acceleration of the object, the initial velocity of the object and the final velocity of the object. While Newton’s three laws have co-operated to help create and improve the study of
Newton-Raphson method is of use when it comes to approximating the root or roots of an equation.
-In order to solve this differential equation you look at it till a solution occurs to you.
Newton on the other had would not allow himself to be usurped by stating that “second inventors have no right. Whether Mr Leibniz found the Method by himself or not is not the Question… We take the proper question to be,… who was the first inventor of the method." In addition, he continued on by stating that "to take away the Right of the first inventor, and divide it between him and that other, would be an Act of Injustice."
Method of doubt is a systematic deduction where all beliefs are rejected, and on the next step they are checked whether they are true with certainty or not before they become knowledge. Father of this method is Renee Descartes. Since Descartes felt that the knowledge wasn’t on solid grounds, he started his search for truth with two tools: doubt and analysis. Starting by doubting everything for finding truth, doubt is pivotal tool for him.
Calculus, the mathematical study of change, can be separated into two departments: differential calculus, and integral calculus. Both are concerned with infinite sequences and series to define a limit. In order to produce this study, inventors and innovators throughout history have been present and necessary. The ancient Greeks, Indians, and Enlightenment thinkers developed the basic elements of calculus by forming ideas and theories, but it was not until the late 17th century that the theories and concepts were being specified. Originally called infinitesimal calculus, meaning to create a solution for calculating objects smaller than any feasible measurement previously known through the use of symbolic manipulation of expressions. Generally accepted, Isaac Newton and Gottfried Leibniz were recognized as the two major inventors and innovators of calculus, but the controversy appeared when both wanted sole credit of the invention of calculus. This paper will display the typical reason of why Newton was the inventor of calculus and Leibniz was the innovator, while both contributed an immense amount of knowledge to the system.
...ke the vanishing determinant, a fixed value of λ = λk, is chosen accordingly. Therefore, at λ = λk, the coefficients of the unknown amplitude Aj in equation (2.12) will become fixed and then it will be possible in obtaining the solution Ajk (the additional subscript k will be used to indicate the correspondence with the particular values of λk). Such a system of equations does not determine the Ajk uniquely but gives their ratios. A convenient mathematical solution designated by the quantities mjk are defined in terms of an arbitrary solution A_jk^' by the formula
The binomial theorem, according to both Haggard et al and Dhand in his lecture (n.d.), is a way of presenting in an equation the expansion of a power of a binomial (x+y)^n but with all the terms; both numerical and literal coefficients are added. The theorem states:
Differential calculus is a subfield of Calculus that focuses on derivates, which are used to describe rates of change that are not constants. The term ‘differential’ comes from the process known as differentiation, which is the process of finding the derivative of a curve. Differential calculus is a major topic covered in calculus. According to Interactive Mathematics, “We use the derivative to determine the maximum and minimum values of particular functions (e.g. cost, strength, amount of material used in a building, profit, loss, etc.).” Not only are derivatives used to determine how to maximize or minimize functions, but they are also used in determining how two related variables are changing over time in relation to each other. Eight different differential rules were established in order to assist with finding the derivative of a function. Those rules include chain rule, the differentiation of the sum and difference of equations, the constant rule, the product rule, the quotient rule, and more. In addition to these differential rules, optimization is an application of differential calculus used today to effectively help with efficiency. Also, partial differentiation and implicit differentiation are subgroups of differential calculus that allow derivatives to be taken to more challenging and difficult formulas. The mean value theorem is applied in differential calculus. This rule basically states that there is at least one tangent line that produces the same slope as the slope made by the endpoints found on a closed interval. Differential calculus began to develop due to Sir Isaac Newton’s biggest problem: navigation at sea. Shipwrecks were frequent all due to the captain being unaware of how the Earth, planets, and stars mov...